Astrophysics and Space Astronomy - Assignment Example

Astronomy Assignment (Order #117442) By JT Thomas 1a) Hotter, more massive stars burn hydrogen more quickly than cooler, less massive stars (see “Main Sequence Stars“). Stars above the turn-off mass leave the main sequence because they have exhausted the hydrogen at their cores (Kaufmann, pp.393-399), leaving a helium core with a hydrogen-burning shell; the core shrinks as the star expands and becomes a red giant. As a red giant, they are cooler and brighter than the main sequence stars. Possibilities for what a massive stars that has left the main sequence may look like include: a red giant, planetary nebula, white dwarf, or black hole (Smith). 1b) Larger mass stars burn hydrogen faster than lower mass stars, so usually it is the larger mass stars that leave the main sequence first. A star’s luminosity is proportional to mass (Danforth). This makes the existence of blue stragglers (which from H-R diagrams, have higher luminosity than other main sequence stars) surprising since they are more massive than many stars that have already left the main sequence 1c) Blue stragglers are possibly the result of two stars merging (see “Star Clusters“), which makes the combined mass hotter and brighter than stars on the main sequence. The more massive hydrogen-depleted star of the merger may form a Roche lobe and dump mass unto the lower-mass star (Danforth), so that it becomes a more massive blue straggler. Another possible cause of blue stragglers is a process by which hydrogen from outer layers of the star becomes mixed into the core, so that core hydrogen burning can continue long after stars of similar mass have become red giants (Kaufmann, p.399). 1d) Stars slightly brighter or redder than the main sequence stars indicate they may be binary stars. Contact binary stars have been found below the turnoff point and slightly redder than the main sequence (Kaluzny et.al.). Main sequence binaries can form a sequence of stars displaced to the red of the main sequence stars (Bailyn and Rubenstein); therefore a large population of binaries may shift the main sequence ridgeline to the red and change the globular cluster luminosity function. 2a) Newton’s version of Kepler’s 3rd law is (see “Kepler‘s Laws of Planetary Motion“): T2 = 4 2 r3 / (MBH G) (for small orbiting mass) Solving for M: MBH = 4 2 r3 / (G T2) Now using r = 1000 AU and T = 15 years: MBH = 4 2 [(1000 AU) (150 E9 m) / (1 AU)]3 / {(6.67 E-11 Nm2/kg2)[( 15 years)(3.156 E7 sec/year)]2} = 8.9 x 1036 kg = 8.9 x 1036 kg * (1 solar mass / 1.99 x 1030 kg) = 4.5 x 106 solar masses So the mass of the Galactic centre black hole is found to be about 4½ million solar masses. 2b) The Schwarzschild radius for a black hole can be found from (Hartle, p.188): Rs = 2 G MBH / c2 where G is Newton’s gravitational constant, MBH is the black hole mass, and c is the speed of light. For the Galactic centre black hole: Rs = 2 (6.67 E-11 Nm2/kg2)(8.9 E36 kg) / (3.00 E8 m/s2) = 1.3 x 1010 m The Schwarzschild radius of a 10 solar mass black hole is: Rs = 2 (6.67 E-11 Nm2/kg2)(10)(1.99 E30 kg) / (3.00 E8 m/s2) =2.95 x 104 m = 29.5 km where we have used the mass of the sun (Halliday and Resnick, p.A5) as 1.99 x 1030 kg. 2c) The tidal force for outside the event horizon (assuming Newtonian gravity): Ftidal = 2 G MBH m h / Rs3 For height = h = 2 meters, and mass = m = 70 kg: Ftidal (at Galactic centre) = 2 (6.67 E-11 Nm2/kg2)(8.9 E36 kg) (70 kg)(2 m) / (1.3 E10 m)3 = 0.076 N Ftidal (10 Msolar black hole)= = 2 (6.67 x 10-11 Nm2/kg2)(10)(1.99 x 1030 kg) (70 kg)(2 m) / (29.5 km)3 = 1.45 x 1010 N You would feel a stronger tidal pull at the 10 solar mass black hole than at the Galactic centre black hole. Force at Earth’s surface on 70 kg person = m g = (70 kg) (9.81 m/s2) = 687 N So the Galactic centre tidal force is 9040 times weaker than the force of gravity at Earth. But the tidal force at the 10 solar mass black hole is approximately 20 million times stronger than the force of gravity at the Earth’s surface. 2d) Using the difference of the tidal force between that on your feet and on your head: Ftidal = G M m /Rs2 - G M m / (Rs + h)2 The tidal force near a 10 solar mass black hole is: Ftidal = (6.67 x 10-11 Nm2/kg2)(10)(1.99 x 1030 kg) (70 kg) / (2.95 x 104 m)2 + - (6.67 x 10-11 Nm2/kg2)(10)(1.99 x 1030 kg) (70 kg) / (2.95 x 104 m + 2 m) 2 = (1.0677 - 1.0675) x 1014 N = 1.45 x 1010 N At the Galactic centre black hole, the tidal force is: Ftidal = (6.67 x 10-11 Nm2/kg2)(8.9 x 1036 kg) (70 kg) / (1.3 x 1010 m)2 + - (6.67 x 10-11 Nm2/kg2)(8.9 x 1036 kg) (70 kg) / (1.3 x 1010 m + 2 m) 2 = 0.07 N where the numbers in the last line were calculated with Derive 5 software. Bibliography Bailyn, Charles D. and Rubenstein, Eric P. “Hubble Space Telescope Observations of the Post- Core Collapse Globular Cluster NGC 6572. II. A Large Main-Sequence Binary Population”. The Astrophysical Journal (1997 Jan 10) 474:701-709. Danforth, Charles. “Blue Stragglers: A Study of Stellar Longevity”. 19 Jan 1998. University of Colorado at Boulder, Center for Astrophysics and Space Astronomy. 22 Mar 2006 http://casa.colorado.edu/~danforth/science/bss/index.html. Halliday, David and Resnick, Robert. Fundamentals of Physics, 3rd Ed. Extended. NewYork: John Wiley & Sons, 1988, p.A5. Hartle, James B. Gravity: An Introduction to Einstein’s General Relativity. San Francisco: Addison Wesley, 2003, p.188. Kaluzny, J.; Kreminski, W.; and Nalezyty, M. “New variable stars in the globular cluster NGC 288”. Astronomy and Astrophysics Supplement Series Vol. 125 (1997 Oct 2): 337-341. Kaufman, William J. III. Universe. New York: W. H. Freeman and Company, 1985. “Kepler’s Laws of Planetary Motion”. 21 Mar 2006.Wikipedia. 22 Mar 2006. http://en.wikipedia.org/wiki/Kepler%27s_laws#Kepler.27s_third_law. “Main Sequence Stars” The Electronic Universe, University of Oregon. 22 Mar 2006 http://zebu.uoregon.edu/~js/ast122/lectures/lec14.html. Smith, Gene. “Gene Smith’s Astronomy Tutorial: Stellar Evolution I - Solar Type Stars” University of California, San Diego. 22 Mar 2006. http://cassfos02.ucsd.edu/public/tutorial/StevI.html. “Star Clusters”, Australian Telescope Outreach and Education. 22 Mar 2006. http://outreach.atnf.csiro.au/education/senior/astrophysics/stellarevolution_cluster s.html. Read More